Listen to Robert Emmerich introduce The Big Apple, a hit song from 1937. Music written by Bob and performed by Tommy Dorsey's Clambake Seven with Bob on piano. Lyrics written by Buddy Bernier and sung by Edythe Wright. Audio provided by Dorothy Emmerich.

GRAY LADY
"Old Gray Lady" dates, at least, from the 1950s. It's an echo of the nickname "the Old Lady of Threadneedle Street," long-used for the Bank of England.

Google Books
17 September 1951, Life magazine, "The Gray Lady Reaches 100" by Meyer Berger, pg. 153, col. 1:
THE Old Gray Lady will celebrate her 100th birthday this Sept. 18. The "lady" is a newspaper -- the New York Times -- regarded by many in the world at large (and all within its own world) as the world's greatest. And newsmen generally hail it as "old" and "gray" by way of acknowledging its traditional special marks: starch conservatism and circumspection.

18 September 1951, New York (NY) Times, pg. 26:
BENJAMIN H. NAMM - Heartiest congratulations upon the 100th anniversary of THE NEW YORK TIMES. May the old gray lady never die and its presses keep rolling along.

August 1958, Harper's Magazine, pg. 28:New York's GRAY OLD LADY
by BEN RATHBUN
(...)
Was this the Times as we once knew it? The fact is, in recent years, that the "gray old lady" has taken to behaving in an uncommonly vivacious way. Among other things, she has trafficked regularly in whimsy, flippant headlines, sprightly writing, and frivolous chit-chat about personalities. Sometimes she has even been known to act like an old gossip.

PAPER OF RECORD
"Paper of Record" or "newspaper of record" was said of other newspapers before the Times, but the usage helped establish the term.

October 26, 1924, New York (NY) Times, pg. 55 ad:The Newspaper of Record. The New York Times is the newspaper of record.

September 18, 1926, New York (NY) Times, pg. 20 (Comments on the New York Times on its 75th Anniversary):
The New York Times is the newspaper of record.

March 12, 1939, New York (NY) Times, pg. 154:
Since this is the paper of record...

April 24, 1958, New York (NY) Times, pg. 30:
Since The New York Times is known throughout the world as a paper of record...

September 23, 1959, New York (NY) Times, pg. 38 (Felix Franfurter):
...rightly characterized as the nation's greatest paper of record.

ALL THE NEWS THAT'S FIT TO PRINT
"All the news that's fit to print" was coined by published Adolph S. Ochs. There was a contest to replace it, but the publisher liked his own creation.

December 4, 1890, New York (NY) Times, pg. 4:
The excellence and interest of The Times as a general newspaper are proverbial. It is its business to print the news, all the news that it is worth anybody's time to read.

October 9, 1896, New York (NY) Times, pg. 7:
All the news that's fit to print.

OLD INDIGESTIBLE
Finally, there's this one:

A Mole's-Eye View of New York
by Democritus Haschid (pseud. - ed.)
Boston, MA: Charles T. Branford Company
1941
Pg. 244: "OLD INDIGESTIBLE" This baptismal was long ago bestowed on The Times by the Subway Ladies, who shun it.